Bevorstehende Vorträge im Oberseminar

Vergangene Vorträge

Lyapunov functions are a well known tool in the theory of dynamical systems to distinguish different regimes of the dynamics. A similar problem appeared in the theory of spacetimes for General Relativity, in the form of time/temporal functions. The theory of Lyapunov functions for cone fields joins both points of view into a single one. In my talk I will introduce Lyapunov functions for cone fields and the main results of the theory. At the end I will comment on new developments in the field. This is joint work with Patrick Bernard (ENS Paris).

I will explain how to construct symbolic dynamics for 3D Reeb flows with non-degenerate periodic orbits when a transversal foliation exists, e.g. via the Hofer-Wysocki-Zehnder holomorphic curve theory. The following dichotomy is a consequence: Either there is a global section or a horse-shoe. This is joint work with Umberto Hryniewicz and Gerhard Knieper.

Anna Florio (IMJ-PRG):Torsion of Conservative Twist Maps on the Annulus

For a C1 diffeomorphism f on the annulus isotopic to the identity, the torsion is the limit of the average rotational velocity of the images of tangent vectors through the differential of f. We consider torsion of conservative twist map on the annulus and we discuss conditions to assure the existence of points with non-zero torsion. As an outcome, we prove that the set of points with non-zero torsion within bounded instability regions has positive Lebesgue measure.

Reeb flows are an important class of dynamical systems which
appear in geometry, symplectic topology and mathematical physics. The
topological entropy is a non-negative number associated to a dynamical
system which quantifies the exponential instability of the system.
Positivity of the topological entropy means that the dynamical system
presents some type of chaotic behavior. In this talk I will explain
how one can extract information about the topological entropy of Reeb
flows from the behavior of invariants from symplectic topology. These
invariants are constructed, following a recipe discovered by Floer, by
studying the spaces of solutions of certain partial differential
equations on a symplectic manifold. The talk will be aimed also at
non-specialists. Our hope is to present to viewers some of the
techniques used to study the dynamics Reeb flows by looking at their
applications to the entropy problem.

Gabriele Benedetti (Universität Heidelberg):Periodic motions of a charged particle in a stationary magnetic field

In this talk, I will present some recent developments about
periodic motions of a charged particle in a magnetic field on a closed
manifold. Using the topology of the manifold and tackling some
interesting analytic issues, one can show that periodic motions exists
for almost every value of the kinetic energy. When the manifold is
two-dimensional and the magnetic field is large with respect to the
kinetic energy much stronger results are known. As a sample, we show
that there exists a countable family of magnetic fields on the
two-torus such that all motions with unit kinetic energy are periodic.

After I describe a Riemannian geometric structure of
somewhat algebraic nature (symmetric spaces), I'll discuss the minimal
entropy conjecture and its relation to the study of other global
Riemannian invariants, including the minimal volume of Gromov, the
length spectrum and the bounded cohomology.